The best-known properties and formulas of partitions

Simple values at zero and infinity

The partition functions and are defined for zero and infinity values of argument by the following rules:

Specific values for specialized variables

The following table represents the values of the partitions and for and some powers of 10:

Analyticity

The partition functions and are non‐analytical functions that are defined only for integers.

Periodicity

The partition functions and do not have periodicity.

Parity and symmetry

The partition functions and do not have symmetry.

Series representations

The partition functions and have the following series representations:

where is a special case of a generalized Kloosterman sum:

Asymptotic series expansions

The partition functions and have the following asymptotic series expansions:

Generating functions

The partition functions and can be represented as the coefficients of their generating functions:

where is the coefficient of the term in the series expansion around of the function , .

Identities

The partition functions and satisfy numerous identities, for example:

Complex characteristics

As real valued functions, the partitions and have the following complex characteristics:

Summation

There exist just a few formulas including finite and infinite summation of partitions, for example:

Inequalities

The partitions and satisfy various inequalities, for example:

Congruence properties

The partitions have the following congruence properties:

Zeros

The and partitions have the following unique zeros: